Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{L}}_\mathbf{1}$$\end{document}-potentials and an Ambartsumian Theorem

In this paper we study Schrödinger operators with absolutely integrable potentials on metric graphs. Uniform bounds—i.e. depending only on the graph and the potential—on the difference between the nth\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^\mathrm{th}$$\end{document} eigenvalues of the Laplace and Schrödinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schrödinger operators with Neumann conditions on an interval. We also extend a previous result relating the spectrum of a Schrödinger operator to the Euler characteristic of the underlying metric graph.


Introduction
Quantum graphs-i.e. Schrödinger operators acting on metric graphs-have become an increasingly important branch of mathematical physics in the last 20 years or so. They serve as models of branched thin networks, e.g. nanotubes, and complex molecules. Apart from applications they also serve as a rich source of objects suitable for mathematical inquiry. In particular Schrödinger operators on graphs exhibit spectral properties both of partial differential and ordinary differential operators.
The aim of this paper is twofold. First we work towards proving a spectral estimate, i.e. a comparison between the spectra of the Laplacian and Schrödinger operators acting on the same metric graph. The main motivation for obtaining such estimates is that the spectrum of the Laplacian is much easier to calculate. We will prove that just as in the case of a single interval the difference between the Laplace and Schrödinger eigenvalues is uniformly bounded, provided the potential is just absolutely integrable. The classical proof of this relied heavily on the explicit formula for the resolvent kernel of the Laplacian [7]. The corresponding kernel for metric graphs is not in general given explicitly and we will instead work with general perturbation theory. The second goal is to extend results-relating the Laplace spectrum to geometric properties of the graph-to Schrödinger operators with L 1 -potentials. In particular we obtain an inverse spectral theorem that generalizes the celebrated theorem of Ambartsumian (see Sect. 6), and show that the Euler characteristic of the underlying graph is reflected in the spectrum of the Schrödinger operator in the case of standard vertex conditions (see (2.1) below).

Spectral Estimates and Inverse Spectral Theory
Inverse spectral theory for the Schrödinger equation in R n has classically had as a goal to determine the potential given a spectrum. To solve the inverse problem for a Schrödinger operator on a metric graph completely one has in general to determine not only the potential but the underlying metric graph and vertex conditions. It appears that this complete inverse problem is rather difficult, especially since the set of spectral data is not obvious. Therefore it seems attractive to start investigations assuming that the metric graph and the vertex conditions are fixed. The simplest graph to be considered is the star graph formed by a finite number of compact edges, and the corresponding inverse problem resembles very much the inverse problem on a single interval, where the potential is determined by two spectra [14,30,[33][34][35]. The case of general trees has also been studied and we have a rather good understanding of the problem [2,8,9,31,37]. The case of graphs with cycles is much more involved-major difficulties are related to the reconstruction of the potential on the cycles. To assure the uniqueness of the potential one may either use the dependence of the spectral data on the magnetic fluxes [21,23], or add extra spectral data like the Dirichlet spectrum [38][39][40][41].
The problem to reconstruct the metric graph has not been addressed in full generality. Topological characteristics of the graph may be reconstructed [19,20]. Assuming that the edge lengths are rationally independent one may even reconstruct the graph using the trace formula [15,18], but under the condition that the potential is zero. Explicit examples of isospectral graphs have been constructed [4,5,15]. Employing the boundary control method one may reconstruct metric trees [2] without assuming zero potential. The problem of reconstructing vertex conditions, as well as the influence of vertex conditions on the solvability of the inverse problem, is even less understood [3]. For example a metric tree is not always reconstructable, e.g. if the vertex conditions are not standard [22]. On the other hand more general vertex conditions may help to solve the inverse problem as it is done in [23,24].
One result in the inverse spectral theory-the Ambartsumian's celebrated theorem from 1929 stating that the spectrum of a (Neumann) Schrödinger operator on an interval coincides with the spectrum of the (Neumann) Laplacian if and only if the potential is zero-is of great importance (see Theorem 6.1). This theorem is rather special, since in order to reconstruct a non-zero potential, knowledge of two spectra is required. Several authors generalized this theorem for the case of metric graphs. The results can be divided into two categories. Some authors proved that if the spectrum of the standard Schrödinger operator on a fixed metric graph Γ coincides with the spectrum of the standard Laplacian on Γ, then the potential is zero. This result was first proved for trees [10,27,29,36] and [30], and then for arbitrary graphs by Davies [12]. It was also noted that the Laplacian on a metric graph Γ is isospectral to the Laplacian on an interval if and only if Γ is formed by a single interval. This can be seen as a geometric version of Ambartsumian's theorem and it is based on the fact that a single interval maximizes the spectral gap for the Laplacian among graphs of fixed total length [13,25,28].
The inverse spectral result that we obtain-Theorem 6.5-is that the spectrum of a finite interval is unique among connected finite compact quantum graphs with standard conditions. More precisely, if the spectrum of a Schrödinger operator on a connected metric graph coincides with the spectrum of the Laplacian with Neumann conditions on an interval, then the graph is the interval, and the potential of the Schrödinger operator is zero.
Here we assume that standard conditions are imposed on the graph. In the case of the graph being a compact interval, standard conditions coincide with Neumann conditions at the end-points. The result can therefore be seen as an extension of Ambartsumian's theorem.

Outline of the Paper
Section 2 contains some fundamentals on quantum graphs. In particular we give some elementary spectral estimates and prove that normalized eigenfunctions of the Laplacian L st 0 are uniformly bounded in the L ∞ norm. Section 3 contains the definition of the operator L st q that we associate with the formal expression L st 0 + q, for q ∈ L 1 (Γ). The operator is defined via quadratic form methods. Section 4 deals with spectral estimates, and we prove that there exists a uniform bound on the difference of eigenvalues of Laplace and Schrödinger operators on metric graphs. This is done by using the Max-Min and Min-Max principles, along with a Sobolev estimate for functions ψ ∈ W 1 2 (Γ). Section 5 gives a result on the zeros of trigonometric polynomials, namely that if the zeros asymptotically tend to the integers, then all the zeros are in fact exactly the integers. By combining this fact with the spectral estimate of Sect. 4 we are able to prove the inverse spectral theorem given in Sect. 6. Section 7 extends a previous result that the Euler characteristic of a graph is reflected in the spectrum of a Schrödinger operator with L ∞ potential to the case of L 1 potentials.

Preliminaries-Basics on Quantum Graphs
A quantum graph is a metric graph equipped with a Schrödinger operator, or more formally, a triple (Γ, L, vc) with Γ a metric graph, L a differential operator, and vc a set of vertex conditions imposed to connect the edges and ensure the self-adjointness of L. In this paper we limit ourselves to compact finite graphs (see below). We give a brief overview of these -for a thorough treatment of the theory of quantum graphs, see for example [6] and [26].
Metric Graphs: A compact finite metric graph is a finite collection of compact intervals of R glued together at the endpoints. More precisely, let, {E n } N n=1 be a finite set of compact intervals, each E n considered as a subset of a separate copy of R, denote the set of endpoints of the intervals. Fix a partition of V into equivalence classes Identifying the endpoints yields a graph with vertices given by the equivalence classes V i . We let n = x 2n − x 2n−1 denote the length of the edge E n and define the total length L of a graph as the sum of its edge lengths: Differential Operators on Graphs: L q acts as a differential operator on each edge separately, in our case − d 2 /dx 2 + q for some real potential function q-subscripts will denote the potential of the operator, and L 0 means that the potential is identically 0, so that L 0 is the Laplacian. The action is in the Hilbert spaces L 2 (E i ) of square integrable functions on the edges E i . The measure on each edge is given naturally by the identification of the edge with an interval. L q then acts on Note that if q ∈ L 1 (Γ) then the formal expression −d 2 /dx 2 + q should be understood as a sum of quadratic forms (see Sect. 3).

Vertex Conditions:
We shall solely study quantum graphs with standard conditions-also known as Kirchoff, Neumann, natural or free conditions. Operators with standard conditions imposed will be written as L st q . Let us first discuss the case of a bounded potential q ∈ L ∞ (Γ), where the domain of the Schrödinger operator can be given explicitly. For any ψ ∈ L 2 (Γ), also qψ ∈ L 2 (Γ), so L q ψ = −(d/dx) 2 ψ + qψ ∈ L 2 (Γ) if and only if ψ ∈ W 2 2 (Γ \ V ). Let V j = {x j1 , . . . , x j k } and let ψ(x ji ) denote the limit ψ(x ji ) = lim x→xj i ψ(x), where the limit is taken over x inside the interval with x ji as an end-point. Standard conditions are then given by imposing the following relations at where ∂ n denotes the normal derivative, i.e.
In other words, ψ is required to be continuous in each vertex V j and the sum of normal derivatives should vanish there. This yields a self-adjoint operator L st q on Γ. We will study L st q for q ∈ L 1 (Γ). This means that in general qψ ∈ L 2 (Γ), so the formal expression L q = −d 2 /dx 2 + q lacks an immediate meaning as a sum of operators from L 2 (Γ) to L 2 (Γ). We will however establish that the perturbation by q is infinitesimally form-bounded (see Proposition 3.2) with respect to the quadratic form of L 0 . More precisely, the quadratic form which are in addition continuous at all vertices. Then the expression L st 0 + q = L st q can be assigned a meaning via the KLMN theorem (see e.g. [32]), as a self-adjoint operator with the same form domain as L st 0 . Note that the domain of the quadratic form only includes the continuity condition from (2.1).

The Spectrum:
We denote the spectrum of L by σ(L). For finite compact graphs the spectrum of L 0 is discrete, for any self-adjoint boundary conditions, and furthermore satisfy Weyl asymptotics. This also holds for L q , for q ∈ L 1 (Γ) (see Sect. 4). The n th eigenvalue, counting multiplicities, of L st q (Γ) is denoted by λ n (L st q ) = λ n (L st q (Γ)). When we write σ(L st q1 (Γ 1 )) = σ(L st q2 (Γ 2 )) we mean not only equality as sets, but also that the multiplicities of all eigenvalues are equal.
For a given finite compact graph of total length L with N edges and M vertices, the following lower and upper estimates can be proven (see "Appendix" for the proof) In particular this estimate implies that the multiplicity of the eigenvalues is uniformly bounded by M + N , since (2.2) implies The above estimate is enough for our purposes, but it may be improved further taking into account the structure of the graph [17].
Another implication of (2.2) is the well-known Weyl asymptotics: We shall also need the following uniform bound on the amplitude of eigenfunctions of the Laplacian on finite metric graphs: where the constant c(Γ) is determined by the graph Γ but is independent of n.
One can find explicit proofs of all the above mentioned formulas (2.2), (2.3), and (2.4) in the "Appendix".

Definition of the Operator
The following well-known Sobolev estimate (a special case of Gagliardo-Nirenberg estimate) is valid for any function ψ ∈ W 1 2 (0, ) on an interval of finite length < ∞ (see proof in the "Appendix") for sufficiently small > 0. The constants, though sufficient for our needs, may be improved if one is willing to sacrifice some elegance in the proof.
In particular any function ψ ∈ W 1 2 (Γ \ V ) will satisfy the estimate (3.1) on each edge E n of Γ, with = n . As the set of edges is finite there is an edge that has minimal length so we obtain a global estimate of |ψ(x)| 2 on Γ: for sufficiently small > 0 and x ∈ Γ.

Proposition 3.2.
Let q ∈ L 1 (Γ), and let Q q be the quadratic form given by and Q L st 0 the quadratic form associated with L st 0 . Then for sufficiently small

Multiplying by q and integrating we obtain
IEOT Schrödinger Operators on Graphs and Geometry II Page 7 of 24 40 The KLMN theorem [32] now lets us conclude that there is a unique bounded from below self-adjoint operator associated with the form Q L st 0 +Q q : Definition 3.1. For q ∈ L 1 (Γ) we denote by L st q (Γ) the operator associated with the form Q L st 0 + Q q . We note that the form domains of L st q and L st 0 coincide, and

Spectral Estimates
We recall the following standard variational theorems, see e.g. [32] for proofs. The lowest eigenvalue will be denoted by λ 0 .

Proposition 4.1. (Min-Max). Let
A be a self-adjoint, bounded from below, operator with discrete spectrum, then the n th eigenvalue of A is given by

Proposition 4.2. (Max-Min). Let
A be a self-adjoint bounded from below, operator with discrete spectrum, then the n th eigenvalue of A is given by In order to apply Propositions 4.1 and 4.2 as we do in the following it is of course required that the spectrum of L st q (Γ) is discrete. For Schrödinger operators with L 1 potentials on finite intervals this is well known, so it is true for finite metric graphs as well since these are just finite-rank perturbationsin the resolvent sense-of the Dirichlet Schrödinger operators (defined by Dirichlet conditions at all vertices), which is nothing else than an orthogonal sum of Dirichlet Schrödinger operators on a collection of finite intervals.
We now proceed to prove the spectral estimate for finite compact graphs, i.e. we show that the difference between the Laplace and the Schrödinger eigenvalues is uniformly bounded: Theorem 4.3. Let Γ be a finite compact metric graph, and let q ∈ L 1 (Γ). Then the difference between the eigenvalues λ n (L st 0 ) and λ n (L st q ) is bounded by a constant, i.e. Similar questions have been discussed in [11] for the case of equilateral metric graphs.
We are going to prove the Theorem using Propositions 4.1 and 4.2. To illustrate the strategy let us first try to derive an upper estimate for λ n (L st q ) using a more naive approach. The quadratic form is It can be estimated from above by where q + is the positive part of the potential q: This step cannot be improved much, since the new estimate coincides with the original one in the case where q is nonnegative. The idea how to proceed is to choose a particular n-dimensional subspace V 0 n . Then the Rayleigh quotient gives not an exact value for λ n−1 (L st q ), but an upper estimate when Proposition 4.1 is used The only reasonable candidate for V 0 n we have at hand is the linear span of the Laplacian eigenfunctions corresponding to the n lowest eigenvalues If q ≡ 0 then this estimate gives the exact value for λ n−1 . Therefore it is natural to split the quadratic form as follows: Then the first quotient is equal to λ n−1 (L st 0 ) and the maximum is attained on If nothing about q is known, then to estimate the second quotient one may use (4.5) IEOT Schrödinger Operators on Graphs and Geometry II Page 9 of 24 40 We need to estimate |u(x)| 2 , provided u = and i.e. we do not get an estimate uniform in n-the estimate grows linearly with n. The reason is the splitting of the quadratic form of L st q into two parts. To obtain the upper bounds we used two intrinsically different vectors: the first term is maximized if u = ψ L st 0 n−1 , while to estimate the second term we used u = ψ This is the reason that the estimate (4.7) is not optimal.
Proof of Theorem 4.3. We divide the proof into two parts deriving upper and lower estimates separately.

Upper Estimate
As before we use the estimate where V 0 n is defined by (4.4). Every function u = n−1 from V 0 n can be written as a sum u = u 1 + u 2 , where Here p is a natural number to be fixed later (independent of n, but depending on Γ and q). Therefore as n increases the first function u 1 will contain an increasing number of terms, while the second function will always be given by a sum of p terms.
From the inequality |u 1 + u 2 | 2 dx ≤ 2 |u 1 | 2 dx + 2 |u 2 | 2 dx and the fact that q + is nonnegative we have That u 1 , u 2 are orthogonal is clear, and from this the orthogonality of u 1 and u 2 also follows: Taking this into account we arrive at . (4.10) To estimate the first form we use (4.5) and the Sobolev estimate (3.2) for max |u(x)|. We get we get The key point is that and p can be chosen in such a way that holds (see (4.14) below).
On the other hand, our naive approach (4.6) can be applied to the second form with the only difference being that the number of eigenfunctions involved is p, not n IEOT Schrödinger Operators on Graphs and Geometry II Page 11 of 24 40 Putting together the obtained estimates in (4.10) and using that u 2 . We would get the desired estimate with C = 2c 2 p q + L1 if we manage to prove that for a certain that may depend on n and p. We use the estimate for Laplacian eigenvalues given in (2.2): Substituting λ n−1 (L st 0 ) with the lower bound and λ n−p−1 (L st 0 ) with the upper and setting = 1/n, we get the following inequality for the left-hand side of (4.14) We see that for any integer p > M + N − 1 + (1 + 2(L/π) 2 ) q + L1 the expression is positive for sufficiently large n and the difference between the eigenvalues possesses the uniform upper estimate: If one is interested in the difference between the eigenvalues for large n only, then the constant C can be taken equal to but this value of C may be too small in order to ensure that (4.16) holds for all n, since proving (4.14) we assumed that n is sufficiently large. The latter assumption does not affect the final result, since for a finite number of eigenvalues (4.16) is always satisfied, but the value of the constant C may be affected.

Lower Estimate
To obtain a lower estimate we are going to use the Max-Min principle (Proposition 4.2). The first step is to notice that Using the subspace V 0 n−1 defined in (4.4) we get Since u is orthogonal to V 0 n−1 it possesses the representation As before let us split the function u = u 1 + u 2

Note two important differences:
• the function u 1 is given by the sum of p terms, where the number p independent of n will be chosen later, so the functions u 1 and u 2 exchange roles compared with the proof of the upper estimate; • the function u 2 is given by an infinite series, not by an increasing number of terms as the function u 1 in the proof of upper estimate.
Using the fact that q − is nonnegative we may split the quadratic form (compare (4.10)) .

(4.18)
Now the function u 1 is given by a finite number of terms and we may similarly to (4.12) estimate To estimate the second form we use (4.5) and the Sobolev estimate (3.2) for max |u(x)| 2 . We get Taking into account we arrive at Summing the estimates (4.19) and (4.21) and taking into account that As before, to prove the desired uniform estimate it is sufficient to show that for large enough n the following expression can be made positive by choosing an appropriate : Again we use (4.15): we substitute λ n+p−1 (L st 0 ) with the lower bound and λ n−1 (L st 0 ) with the upper. As before we choose = 1/n, so the left-hand side of (4.22) becomes If p > M + N − 1 + (1 + 2(L/π) 2 ) q − L1 , then for sufficiently large n the expression is positive, hence the following lower estimate holds (4.23) where the exact value of C is determined by the difference between the first few eigenvalues as described above. Proof. This is an immediate consequence of (2.3) and Theorem 4.3.
More importantly we may now show that the effect of an L 1 -perturbation on the eigenvalues will tend to zero in n in the scale of square roots. This step is critical in the proof of Theorem 6.5.
n , for some constant C 0 ∈ R. In particular, |k n,q − k n,0 | → 0, n → ∞. (4.24) Proof. Since the eigenvalues satisfy Weyl asymptotics that depend only on the length of Γ we have that k n,q , k n,0 ≥ nD for some constant D and sufficiently large n. We have for some constant C 0 ∈ R.

On the Zeros of Trigonometric Polynomials
In this section we recall the secular equation for the spectrum σ(L st 0 (Γ)): it is given as the squares of the zeros of a trigonometric polynomial. We then prove that if the zeros k m of such a finite trigonometric polynomial with constant coefficients are close to a certain equispaced sequence, i.e. satisfy |k m − mπ/L| → 0 then in fact k m = mπ/L for all m (Theorem 5.2). From this we then prove the geometric Ambartsumian Theorem 6.5.
a n e iωnk with k-independent coefficients a n ∈ C and ω n ∈ R: λ n (L st 0 ) = k 2 n,0 if and only if p(k n,0 ) = 0.
Proof. We sketch the proof: for each non-zero eigenvalue the corresponding eigenfunction is edge-wise just a sum of sine and cosine functions: ψ(x) = a j cos(kx) + b j sin(kx), x ∈ E j . The solutions have to satisfy the vertex conditions. Continuity can at each V be written as where k has been factored out. This yields an additional M equations, so that we in total have 2N equations, which may be written in the form: with c a vector of the coefficients a i , b i and T (k) a matrix with trigonometric entries depending on k. A real number λ = k 2 is an eigenvalue of L st 0 if and only if k is a root of the trigonometric polynomial det T (k) = 0. We refer to [6,18] for details. See also eg. [15] and [26].
Note that it is crucial that the vertex conditions were standard: in the case of more general vertex conditions the secular equation is given by a quasipolynomial instead of the trigonometric polynomial. where d(·, ·) denotes the metric on the J-torus. Taking a sequence i → 0 we may chose i 1 ( i ) = I( i ) and in each step i 2 ( i ) > I( i ) so large that the difference is an increasing sequence. It follows that Remark 5.1. Note that in the special case where 2π, ω 1 , . . . , ω J are rationally independent the classical theorem of Kronecker (see e.g. [16]) can be used. as n → ∞. The limit follows from the choice of m n and the fact that γ (r+mn) tends to 0. The above calculation shows that p(r) = 0. But p(r) = 0 and k r − r → 0 together imply that k r = r, since even a single extra zero would make the asymptotic behaviour impossible.

Remark 5.2.
It is not important in the above theorem that k n tends to the integers. A scaling argument allows one to extend it to the case where k n are close to integer-multiples of an arbitrary real number.

An Ambartsumian Theorem
With the result of the previous two sections we are now in a position to prove an inverse spectral theorem that may be seen as a generalization of Ambartsumian's classical theorem. For the proof we recall the classical result as well as its geometric version for Laplacians.
The following theorem has been a source of inspiration for researchers in inverse problems for almost a century. In the original article [1] it was assumed that the potential is continuous, but the result holds even if q ∈ L 1 . We adjusted the formulation to our notations.  [10,27,29,36] and [30], but the most general version was given by E.B. Davies [12] (Davies proved it for q ∈ L ∞ (Γ) but noted that this condition surely can be weakened).
Proposition 6.2. (following [12]) Let L st q (Γ) and L st 0 (Γ) be the standard Schrödinger and Laplace operators on a connected finite compact metric graph Γ. Assume that the potential q is absolutely integrable. Then the eigenvalues of the two operators coincide if and only if the potential q is equal to zero almost everywhere.
The second theorem is a geometric version of Ambartsumian theorem for standard Laplacians. We start by recalling the result that among graphs with fixed total length the spectral gap is minimized by the single interval [13,28] and [25]: The assumptions of the theorem can be weakened, since to ensure that Γ and I have the same total length it is enough to check the asymptiotics.
Our goal is to prove that if the spectrum of a Schrödinger operator on a metric graph coincides with the spectrum of the Laplacian on an interval then the graph coincides with the interval and the potential is zero. This statement cannot be proven as a simple combination of the above mentioned results (Proposition 6.2 and Theorem 6.4). The main difficulty is to show that the graph coincides with the interval. Theorem 6.4 cannot be applied directly, since it requires q ≡ 0. The spectral estimate (4.1) implies that Hence the square roots of the eigenvalues of L st 0 (Γ) satisfy But k n are given as the zeros of a trigonometric polynomial (Theorem 5.1).
Hence the spectrum of the Laplacian on Γ given by zeroes of the trigonometric polynomial is asymptotically close to a set of equidistant points and Theorem 5.2 can be applied. We conclude that in fact k n (L st 0 (Γ)) = πn/L, for all n. This means that the spectrum of the Laplacian on Γ coincides with the spectrum of the Laplacian on an intreval. We may finally apply the geometric version of Ambartsumian theorem for Laplacians (Theorem 6.4) to conclude that the graph Γ coincides with the interval I. But then Ambartsumian's classical theorem (Proposition 6.1) implies that the potential is identically equal to zero q ≡ 0.
Proving our main theorem we have shown that in order to ensure that the graph Γ coincides with the interval, it is enough to require that the spectrum of the Schrödinger operator is close to the spectrum of the Laplacian on an interval. In fact we proved the following theorem: Theorem 6.6. Let L st q (Γ) be standard Schrödinger operator with L 1 -potential on a finite compact connected metric graph Γ. Then Γ = I if

Euler Characteristic
The spectral estimate (4.1) allows us to extend a previous result of one of the authors regarding the Euler characteristic χ = M − N .
Theorem 7.1. Let Γ be a finite compact metric graph and q ∈ L 1 (Γ). Then the Euler characteristic χ(Γ) is uniquely determined by the spectrum σ(L st q (Γ)), and can be calculated as the limit with the convention that λ n = 0 implies Proof. In [19] formula (7.1) was proven for the case of zero potential. In other words formula (7.1) gives the Euler characteristics if one substitutes the eigenvalues λ n (L st 0 ). We have shown that the difference between λ n (L st q (Γ)) and λ n (L st 0 (Γ)) is uniformly bounded and therefore Taking into account Weyls asymptotics one may apply Lemma 2 from [20] to conclude that formula (7.1) gives the same result independently of whether the eigenvalues of the standard Laplacian L st 0 (Γ) or of the standard Schrödinger operator L st q (Γ) are substituted.
Note that proving the theorem we assumed that the vertex conditions are standard; this assumption in general cannot be removed.

Discussion
It would be interesting to understand under which assumptions a similar result holds for other than the standard vertex conditions that are assumed throughout in this paper. The cases, where the vertex conditions are such that the coefficients of the trigonometric polynomial given by Theorem 5.1 are k independent, could potentially be dealt with in a similar way as in this paper. The bounds on the differences of the eigenvalues obtained in Theorem 4.3 can be established for other choices of vertex conditions, but Theorem 6.4 in general may fail.
Proof of formula (2.2). For a bounded from below self-adjoint operator A with discrete spectrum, define the eigenvalue counting function The standard Laplacian is positive (hence the lower estimate in (2.2) is interesting only if n > M) therefore when calculating the eigenvalue counting function we assume that λ ≥ 0.
Consider first the Laplace operator L D 0 on a single interval I of length with Dirichlet conditions at the end-points. The eigenvalues are λ n = πn 2 , n = 1, 2, . . . So the eigenvalue counting function for λ ≥ 0 is in fact given by where square-brackets denote the integer-part of the argument. Returning to Γ we note that if we impose Dirichlet conditions on the vertices of Γ-denote the operator by L D 0 (Γ)-then we really just have a decoupled set of intervals and therefore the set of eigenvalues is just the union of the eigenvalues for each interval (counting multiplicities). Therefore the corresponding counting function E L D 0 (Γ) is given by since taking integer parts may only decrease the value. First adding and then taking integer parts may-compared to adding the integer parts-at most raise the value by the number of terms −1, i.e. the number of edges −1, so conversely we also have √ λ π L − N + 1 ≤ E L D 0 (Γ) (λ). Since functions in the domain of L D 0 and L st 0 are continuous, to determine the rank of the resolvent difference, we need to determine dim ker − d 2 dx 2 − λ on continuous functions. Prescribing values u j at each vertex V j in Γ, we see that a unique solution to this boundary value problem is given as follows: on an edge E n = [x 2n−1 , x 2n ] between V i and V j , set for k 2 = λ. Then u is continous on Γ and solves (−d 2 /dx 2 − λ)u = 0, so dim ker(−d 2 /dx 2 − λ) ≤ M . Therefore we have The lower estimate estimate (A.2) can be modified in a similar way, but instead we shall take into account that L st 0 (Γ) ≤ L D 0 (Γ). Really Dirichlet conditions in particular imply the continuity of functions in the domain of the quadratic form. Passing to standard conditions means weakening the conditions on functions in the domain of the quadratic form, since now only continuity is required at the vertices. Therefore the domain of the quadratic form Q L st 0 is larger than that of Q L D 0 , so by the Min-Max principle (see Propo- Setting λ = π 2 L 2 n 2 we obtain n − N + 1 ≤ E L st 0 (Γ) π 2 L 2 n 2 ≤ n + M, so λ n−N +1 ≤ π 2 L 2 n 2 ≤ λ n+M .
Setting n = n + M we get λ n ≥ π 2 L 2 (n − M ) 2 and similarly we find λ n ≤  For < we have 1/ > 1/ and the claim follows.